Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5

Percentage Accurate: 60.3% → 98.5%

Time: 15.2s

Alternatives: 8

Speedup: 8.9×

Specification

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

\[\begin{array}{l} \\ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Initial Program: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Alternative 1: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi}{\frac{alphax}{alphay}}\right)} \cdot \left(alphay \cdot \left(-alphax\right)\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (/
   (log1p (- u0))
   (fma (/ sin2phi alphay) alphax (/ cos2phi (/ alphax alphay))))
  (* alphay (- alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (log1pf(-u0) / fmaf((sin2phi / alphay), alphax, (cos2phi / (alphax / alphay)))) * (alphay * -alphax);
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(log1p(Float32(-u0)) / fma(Float32(sin2phi / alphay), alphax, Float32(cos2phi / Float32(alphax / alphay)))) * Float32(alphay * Float32(-alphax)))
end

Derivation

1. Initial program 59.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. sub-neg 59.1%

      \[\leadsto \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. log1p-def 98.3%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Step-by-step derivation

   1. associate-/r* 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   2. associate-/r* 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   3. frac-add 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]

5. Applied egg-rr 98.2%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]
6. Step-by-step derivation

   1. *-commutative 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{alphay \cdot \frac{cos2phi}{alphax}} + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}} \]

   2. *-commutative 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{alphay \cdot \frac{cos2phi}{alphax} + \color{blue}{\frac{sin2phi}{alphay} \cdot alphax}}{alphax \cdot alphay}} \]

   3. +-commutative 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax + alphay \cdot \frac{cos2phi}{alphax}}}{alphax \cdot alphay}} \]

   4. fma-def 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}}{alphax \cdot alphay}} \]

   5. *-commutative 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi}{alphax} \cdot alphay}\right)}{alphax \cdot alphay}} \]

   6. associate-*l/ 98.1%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi \cdot alphay}{alphax}}\right)}{alphax \cdot alphay}} \]

7. Simplified 98.1%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi \cdot alphay}{alphax}\right)}{alphax \cdot alphay}}} \]
8. Step-by-step derivation

   1. associate-/r/ 98.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi \cdot alphay}{alphax}\right)} \cdot \left(alphax \cdot alphay\right)} \]

   2. associate-/l* 98.7%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi}{\frac{alphax}{alphay}}}\right)} \cdot \left(alphax \cdot alphay\right) \]

   3. *-commutative 98.7%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi}{\frac{alphax}{alphay}}\right)} \cdot \color{blue}{\left(alphay \cdot alphax\right)} \]

9. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi}{\frac{alphax}{alphay}}\right)} \cdot \left(alphay \cdot alphax\right)} \]

10. Final simplification 98.7%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi}{\frac{alphax}{alphay}}\right)} \cdot \left(alphay \cdot \left(-alphax\right)\right) \]

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + sin2phi \cdot {alphay}^{-2}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (/ cos2phi (* alphax alphax)) (* sin2phi (pow alphay -2.0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((cos2phi / (alphax * alphax)) + (sin2phi * powf(alphay, -2.0f)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi * (alphay ^ Float32(-2.0)))))
end

Derivation

1. Initial program 59.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. sqr-neg 59.1%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{\left(-alphay\right) \cdot \left(-alphay\right)}}} \]

   2. sub-neg 59.1%

      \[\leadsto \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\left(-alphay\right) \cdot \left(-alphay\right)}} \]

   3. log1p-def 98.3%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\left(-alphay\right) \cdot \left(-alphay\right)}} \]

   4. sqr-neg 98.3%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. associate-/r* 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}} \]
4. Step-by-step derivation

   1. associate-/r* 98.3%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

   2. clear-num 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}} \]

   3. associate-/r/ 98.3%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay \cdot alphay} \cdot sin2phi}} \]

   4. pow2 98.3%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{\color{blue}{{alphay}^{2}}} \cdot sin2phi} \]

   5. pow-flip 98.5%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{{alphay}^{\left(-2\right)}} \cdot sin2phi} \]

   6. metadata-eval 98.5%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + {alphay}^{\color{blue}{-2}} \cdot sin2phi} \]

5. Applied egg-rr 98.5%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{{alphay}^{-2} \cdot sin2phi}} \]

6. Final simplification 98.5%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + sin2phi \cdot {alphay}^{-2}} \]

Alternative 3: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay} \cdot alphax\\ \mathbf{if}\;sin2phi \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphay\right)}{t_0 + \frac{alphay}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{t_0}{alphay \cdot alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* (/ sin2phi alphay) alphax)))
   (if (<= sin2phi 4.999999980020986e-12)
     (/ (* alphax (* u0 alphay)) (+ t_0 (/ alphay (/ alphax cos2phi))))
     (/ (- (log1p (- u0))) (/ t_0 (* alphay alphax))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / alphay) * alphax;
	float tmp;
	if (sin2phi <= 4.999999980020986e-12f) {
		tmp = (alphax * (u0 * alphay)) / (t_0 + (alphay / (alphax / cos2phi)));
	} else {
		tmp = -log1pf(-u0) / (t_0 / (alphay * alphax));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / alphay) * alphax)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999980020986e-12))
		tmp = Float32(Float32(alphax * Float32(u0 * alphay)) / Float32(t_0 + Float32(alphay / Float32(alphax / cos2phi))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / Float32(t_0 / Float32(alphay * alphax)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999998e-12

   1. Initial program 50.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. sub-neg 50.3%

         \[\leadsto \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. log1p-def 98.4%

         \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/r* 98.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

      2. associate-/r* 98.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      3. frac-add 98.4%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]
   6. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{alphay \cdot \frac{cos2phi}{alphax}} + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}} \]

      2. *-commutative 98.4%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{alphay \cdot \frac{cos2phi}{alphax} + \color{blue}{\frac{sin2phi}{alphay} \cdot alphax}}{alphax \cdot alphay}} \]

      3. +-commutative 98.4%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax + alphay \cdot \frac{cos2phi}{alphax}}}{alphax \cdot alphay}} \]

      4. fma-def 98.5%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}}{alphax \cdot alphay}} \]

      5. *-commutative 98.5%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi}{alphax} \cdot alphay}\right)}{alphax \cdot alphay}} \]

      6. associate-*l/ 98.2%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi \cdot alphay}{alphax}}\right)}{alphax \cdot alphay}} \]

   7. Simplified 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi \cdot alphay}{alphax}\right)}{alphax \cdot alphay}}} \]

   8. Taylor expanded in u0 around 0 76.7%

      \[\leadsto \color{blue}{\frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}} \]
   9. Step-by-step derivation

      1. *-commutative 76.7%

         \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{\color{blue}{sin2phi \cdot alphax}}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} \]

      2. associate-*l/ 76.7%

         \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax} + \frac{alphay \cdot cos2phi}{alphax}} \]

      3. *-commutative 76.7%

         \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\color{blue}{alphax \cdot \frac{sin2phi}{alphay}} + \frac{alphay \cdot cos2phi}{alphax}} \]

      4. associate-/l* 77.0%

         \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{alphax \cdot \frac{sin2phi}{alphay} + \color{blue}{\frac{alphay}{\frac{alphax}{cos2phi}}}} \]

   10. Simplified 77.0%

       \[\leadsto \color{blue}{\frac{alphax \cdot \left(alphay \cdot u0\right)}{alphax \cdot \frac{sin2phi}{alphay} + \frac{alphay}{\frac{alphax}{cos2phi}}}} \]

   if 4.99999998e-12 < sin2phi

   1. Initial program 63.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. sub-neg 63.2%

         \[\leadsto \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. log1p-def 98.3%

         \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/r* 98.3%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

      2. associate-/r* 98.3%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      3. frac-add 98.0%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]

   5. Applied egg-rr 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]
   6. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{alphay \cdot \frac{cos2phi}{alphax}} + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}} \]

      2. *-commutative 98.0%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{alphay \cdot \frac{cos2phi}{alphax} + \color{blue}{\frac{sin2phi}{alphay} \cdot alphax}}{alphax \cdot alphay}} \]

      3. +-commutative 98.0%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax + alphay \cdot \frac{cos2phi}{alphax}}}{alphax \cdot alphay}} \]

      4. fma-def 98.0%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}}{alphax \cdot alphay}} \]

      5. *-commutative 98.0%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi}{alphax} \cdot alphay}\right)}{alphax \cdot alphay}} \]

      6. associate-*l/ 98.0%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi \cdot alphay}{alphax}}\right)}{alphax \cdot alphay}} \]

   7. Simplified 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi \cdot alphay}{alphax}\right)}{alphax \cdot alphay}}} \]

   8. Taylor expanded in sin2phi around inf 95.5%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{alphax \cdot sin2phi}{alphay}}}{alphax \cdot alphay}} \]
   9. Step-by-step derivation

      1. *-commutative 95.5%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{\color{blue}{sin2phi \cdot alphax}}{alphay}}{alphax \cdot alphay}} \]

      2. associate-*l/ 95.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax}}{alphax \cdot alphay}} \]

      3. *-commutative 95.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot alphay}} \]

   10. Simplified 95.6%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{sin2phi}{alphay} \cdot alphax + \frac{alphay}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay} \cdot alphax}{alphay \cdot alphax}}\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. sub-neg 59.1%

      \[\leadsto \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. log1p-def 98.3%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

4. Final simplification 98.3%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

Alternative 5: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 59.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. sqr-neg 59.1%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{\left(-alphay\right) \cdot \left(-alphay\right)}}} \]

   2. sub-neg 59.1%

      \[\leadsto \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\left(-alphay\right) \cdot \left(-alphay\right)}} \]

   3. log1p-def 98.3%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\left(-alphay\right) \cdot \left(-alphay\right)}} \]

   4. sqr-neg 98.3%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. associate-/r* 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}} \]

4. Final simplification 98.4%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

Alternative 6: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{sin2phi}{alphay} \cdot alphax + \frac{alphay}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999980020986e-12)
   (/
    (* alphax (* u0 alphay))
    (+ (* (/ sin2phi alphay) alphax) (/ alphay (/ alphax cos2phi))))
   (/ (- (pow alphay 2.0)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999980020986e-12f) {
		tmp = (alphax * (u0 * alphay)) / (((sin2phi / alphay) * alphax) + (alphay / (alphax / cos2phi)));
	} else {
		tmp = -powf(alphay, 2.0f) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 4.999999980020986e-12) then
        tmp = (alphax * (u0 * alphay)) / (((sin2phi / alphay) * alphax) + (alphay / (alphax / cos2phi)))
    else
        tmp = -(alphay ** 2.0e0) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999980020986e-12))
		tmp = Float32(Float32(alphax * Float32(u0 * alphay)) / Float32(Float32(Float32(sin2phi / alphay) * alphax) + Float32(alphay / Float32(alphax / cos2phi))));
	else
		tmp = Float32(Float32(-(alphay ^ Float32(2.0))) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.999999980020986e-12))
		tmp = (alphax * (u0 * alphay)) / (((sin2phi / alphay) * alphax) + (alphay / (alphax / cos2phi)));
	else
		tmp = -(alphay ^ single(2.0)) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999998e-12

   1. Initial program 50.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. sub-neg 50.3%

         \[\leadsto \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. log1p-def 98.4%

         \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/r* 98.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

      2. associate-/r* 98.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      3. frac-add 98.4%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]
   6. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{alphay \cdot \frac{cos2phi}{alphax}} + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}} \]

      2. *-commutative 98.4%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{alphay \cdot \frac{cos2phi}{alphax} + \color{blue}{\frac{sin2phi}{alphay} \cdot alphax}}{alphax \cdot alphay}} \]

      3. +-commutative 98.4%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax + alphay \cdot \frac{cos2phi}{alphax}}}{alphax \cdot alphay}} \]

      4. fma-def 98.5%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}}{alphax \cdot alphay}} \]

      5. *-commutative 98.5%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi}{alphax} \cdot alphay}\right)}{alphax \cdot alphay}} \]

      6. associate-*l/ 98.2%

         \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi \cdot alphay}{alphax}}\right)}{alphax \cdot alphay}} \]

   7. Simplified 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi \cdot alphay}{alphax}\right)}{alphax \cdot alphay}}} \]

   8. Taylor expanded in u0 around 0 76.7%

      \[\leadsto \color{blue}{\frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}} \]
   9. Step-by-step derivation

      1. *-commutative 76.7%

         \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{\color{blue}{sin2phi \cdot alphax}}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} \]

      2. associate-*l/ 76.7%

         \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax} + \frac{alphay \cdot cos2phi}{alphax}} \]

      3. *-commutative 76.7%

         \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\color{blue}{alphax \cdot \frac{sin2phi}{alphay}} + \frac{alphay \cdot cos2phi}{alphax}} \]

      4. associate-/l* 77.0%

         \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{alphax \cdot \frac{sin2phi}{alphay} + \color{blue}{\frac{alphay}{\frac{alphax}{cos2phi}}}} \]

   10. Simplified 77.0%

       \[\leadsto \color{blue}{\frac{alphax \cdot \left(alphay \cdot u0\right)}{alphax \cdot \frac{sin2phi}{alphay} + \frac{alphay}{\frac{alphax}{cos2phi}}}} \]

   if 4.99999998e-12 < sin2phi

   1. Initial program 63.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. associate-/r* 63.2%

         \[\leadsto \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   3. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   4. Taylor expanded in cos2phi around 0 62.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   5. Step-by-step derivation

      1. mul-1-neg 62.4%

         \[\leadsto \color{blue}{-\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

      2. associate-/l* 62.3%

         \[\leadsto -\color{blue}{\frac{{alphay}^{2}}{\frac{sin2phi}{\log \left(1 - u0\right)}}} \]

      3. distribute-neg-frac 62.3%

         \[\leadsto \color{blue}{\frac{-{alphay}^{2}}{\frac{sin2phi}{\log \left(1 - u0\right)}}} \]

      4. sub-neg 62.3%

         \[\leadsto \frac{-{alphay}^{2}}{\frac{sin2phi}{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}} \]

      5. mul-1-neg 62.3%

         \[\leadsto \frac{-{alphay}^{2}}{\frac{sin2phi}{\log \left(1 + \color{blue}{-1 \cdot u0}\right)}} \]

      6. log1p-def 96.0%

         \[\leadsto \frac{-{alphay}^{2}}{\frac{sin2phi}{\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)}}} \]

      7. mul-1-neg 96.0%

         \[\leadsto \frac{-{alphay}^{2}}{\frac{sin2phi}{\mathsf{log1p}\left(\color{blue}{-u0}\right)}} \]

   6. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{-{alphay}^{2}}{\frac{sin2phi}{\mathsf{log1p}\left(-u0\right)}}} \]

   7. Taylor expanded in u0 around 0 87.9%

      \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
   8. Step-by-step derivation

      1. +-commutative 87.9%

         \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{0.5 \cdot sin2phi + -1 \cdot \frac{sin2phi}{u0}}} \]

      2. mul-1-neg 87.9%

         \[\leadsto \frac{-{alphay}^{2}}{0.5 \cdot sin2phi + \color{blue}{\left(-\frac{sin2phi}{u0}\right)}} \]

      3. unsub-neg 87.9%

         \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{0.5 \cdot sin2phi - \frac{sin2phi}{u0}}} \]

      4. *-commutative 87.9%

         \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{sin2phi \cdot 0.5} - \frac{sin2phi}{u0}} \]

   9. Simplified 87.9%

      \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{sin2phi}{alphay} \cdot alphax + \frac{alphay}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 7: 76.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{sin2phi}{alphay} \cdot alphax + \frac{alphay}{\frac{alphax}{cos2phi}}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* alphax (* u0 alphay))
  (+ (* (/ sin2phi alphay) alphax) (/ alphay (/ alphax cos2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphax * (u0 * alphay)) / (((sin2phi / alphay) * alphax) + (alphay / (alphax / cos2phi)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (alphax * (u0 * alphay)) / (((sin2phi / alphay) * alphax) + (alphay / (alphax / cos2phi)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(alphax * Float32(u0 * alphay)) / Float32(Float32(Float32(sin2phi / alphay) * alphax) + Float32(alphay / Float32(alphax / cos2phi))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (alphax * (u0 * alphay)) / (((sin2phi / alphay) * alphax) + (alphay / (alphax / cos2phi)));
end

Derivation

1. Initial program 59.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. sub-neg 59.1%

      \[\leadsto \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. log1p-def 98.3%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Step-by-step derivation

   1. associate-/r* 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   2. associate-/r* 98.4%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   3. frac-add 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]

5. Applied egg-rr 98.2%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}}} \]
6. Step-by-step derivation

   1. *-commutative 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{alphay \cdot \frac{cos2phi}{alphax}} + alphax \cdot \frac{sin2phi}{alphay}}{alphax \cdot alphay}} \]

   2. *-commutative 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{alphay \cdot \frac{cos2phi}{alphax} + \color{blue}{\frac{sin2phi}{alphay} \cdot alphax}}{alphax \cdot alphay}} \]

   3. +-commutative 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax + alphay \cdot \frac{cos2phi}{alphax}}}{alphax \cdot alphay}} \]

   4. fma-def 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}}{alphax \cdot alphay}} \]

   5. *-commutative 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi}{alphax} \cdot alphay}\right)}{alphax \cdot alphay}} \]

   6. associate-*l/ 98.1%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \color{blue}{\frac{cos2phi \cdot alphay}{alphax}}\right)}{alphax \cdot alphay}} \]

7. Simplified 98.1%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, \frac{cos2phi \cdot alphay}{alphax}\right)}{alphax \cdot alphay}}} \]

8. Taylor expanded in u0 around 0 76.8%

   \[\leadsto \color{blue}{\frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}} \]
9. Step-by-step derivation

   1. *-commutative 76.8%

      \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{\color{blue}{sin2phi \cdot alphax}}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} \]

   2. associate-*l/ 76.7%

      \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\color{blue}{\frac{sin2phi}{alphay} \cdot alphax} + \frac{alphay \cdot cos2phi}{alphax}} \]

   3. *-commutative 76.7%

      \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{\color{blue}{alphax \cdot \frac{sin2phi}{alphay}} + \frac{alphay \cdot cos2phi}{alphax}} \]

   4. associate-/l* 76.8%

      \[\leadsto \frac{alphax \cdot \left(alphay \cdot u0\right)}{alphax \cdot \frac{sin2phi}{alphay} + \color{blue}{\frac{alphay}{\frac{alphax}{cos2phi}}}} \]

10. Simplified 76.8%

    \[\leadsto \color{blue}{\frac{alphax \cdot \left(alphay \cdot u0\right)}{alphax \cdot \frac{sin2phi}{alphay} + \frac{alphay}{\frac{alphax}{cos2phi}}}} \]

11. Final simplification 76.8%

    \[\leadsto \frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{sin2phi}{alphay} \cdot alphax + \frac{alphay}{\frac{alphax}{cos2phi}}} \]

Alternative 8: 76.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 59.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. associate-/r* 59.1%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

3. Simplified 59.1%

   \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}} \]

4. Taylor expanded in u0 around 0 76.7%

   \[\leadsto \frac{-\color{blue}{-1 \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
5. Step-by-step derivation

   1. mul-1-neg 76.7%

      \[\leadsto \frac{-\color{blue}{\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

6. Simplified 76.7%

   \[\leadsto \frac{-\color{blue}{\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

7. Final simplification 76.7%

   \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

Reproduce

herbie shell --seed 2023312 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (and (and (and (and (<= 0.0001 alphax) (<= alphax 1.0)) (and (<= 0.0001 alphay) (<= alphay 1.0))) (and (<= 2.328306437e-10 u0) (<= u0 1.0))) (and (<= 0.0 cos2phi) (<= cos2phi 1.0))) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))